Part 2
- A model for the visual cortex due to Petitot-Citti and Sarti.
- Geodesics on the bundle of directions of the plane.
- Inpaiting via the sub-elliptic diffusion equation.

March 25
1:10 p.m.
Fields Institute, Room 210

Gunay Dogan, National Institute of Standards and Technology
(NIST)Variational Algorithms for Fast and Robust Shape Reconstruction in
Imaging

In many image processing problems, the end goal or an essential ingredient
is the detection and reconstruction of a shape or a set of shapes
(curves in 2d or surfaces in 3d) from the given data. Examples of
such problems are image segmentation, surface regularization, geometric
interpolation of data point clouds and multiview stereo reconstruction.
Often these tasks can be expressed as energy minimization problems
in which the free variable is the sought shape. In this work, we adopt
a continuous shape optimization approach to such problems and develop
a Lagrangian minimization framework that can handle a diverse collection
of such shape energies. We discretize the continuous formulation with
the finite element method. By formulating well-behaved gradient descent
velocities for shape evolutions and by employing spatial adaptivity,
we obtain an efficient and robust shape reconstruction algorithm.
We demonstrate the effectiveness of our method with several examples
from image processing.

March 18
1:10 p.m.
Fields Institute, Stewart Library

Peter Gibson, York UniversityStable Polynomials and the Identification of Minimum-phase Preserving
Operators

It is claimed by seismologists that causal signals in the form of
elastic waves transmitted through layers of rock end up being minimum
phase, meaning that their energy is maximally concentrated at the
front. We analyze the mathematical implications of this claim. It
turns out that the underlying operators are limited to a precisely
defined class which can be identified using specific test signals.
The key insight comes from the theory of stable polynomials and what
are known as Polya-Schur problems

We discuss the following nonholonomic version of the classical Moser
theorem: given a bracket-generating distribution on a connected compact
manifold (possibly with boundary), two volume forms of equal total
volume can be isotoped by the flow of a vector field tangent to this
distribution. The subriemannian heat equation turns out to be a gradient
flow on the "nonholonomic" Wasserstein space with the potential
given by the Boltzmann relative entropy functional. This is joint
work with Paul Lee.

Friday, Dec. 14
1:10
Bahen 6183

Jeff Calder, University of Michigan
The circular area signature for graphs of periodic functions

The representation of curves by integral invariants is an important
step in shape recognition and classification in the computer vision
community. Integral invariants are preferred over their differential
counterparts due to their robustness with respect to noise. However,
in contrast to differential invariants, it is currently unknown whether
integral invariants offer unique representations of curves. An important
example is the circular area signature, which is of particular interest
due to its relation to curvature. The circular area signature is computed
by placing a ball of fixed radius at each point on the curve and measuring
the area of the portion of the ball that lies inside the region bounded
by the curve. In this talk, I will sketch the proof of a new uniqueness
result for the circular area signature for graphs of periodic functions,
and discuss the difficulties with obtaining a similar result for simple
closed curves.

Friday, Nov. 16
1:10
Bahen 6183

Allen Tannenbaum, Comprehensive Cancer Center/ECE, UABMonge-Kantorovich for Problems in Signal Processing and Control

Monge-Kantorovich (also known as Optimal Mass Transport) is an important
problem with applications for numerous disciplines including econometrics,
fluid dynamics, automatic control, transportation, statistical physics,
shape optimization, expert systems, and meteorology. In this talk,
we will discuss certain computational approaches to this problem and
their applications in signal and image processing. Further via gradient
flows defined by the Wasserstein 2-metric from optimal transport theory,
we will relate some key concepts in information theory with mass transport
that impacts problems in control, statistical estimation, and image
analysis. A goal of this program is to understand the apparent relationship
between mass transport, conservation laws, entropy functionals, on
one hand and probability and power distributions and related metrics
on the other.

In recent work, Boscain et al. have shown how sub-Riemannian ideas
can be used in imaging applications. In particular, a hypoelliptic
heat equation on SE(2) and its corresponding heat kernel lead to an
image diffusion algorithm which implements image inpainting. Heat
kernels are of independent interest as well, since they encode geometric
information. In particular, the sub-Riemannian distance can be recovered
from the short-time behaviour of the heat kernel, just as in Riemannian
geometry. In this talk, we will describe recent work on computing
the short-time behaviour of hypoelliptic heat kernels on unimodular
Lie groups, and we will illustrate it for the case of the Cartan,
and Engel sub-riemannian geometries.

Stephen Marsland, Massey University, New ZealandShape Metrics, Correspondence, and the Role of the Connection

Metrics on shape space are used to describe deformations that take
one shape to another, and to determine a distance between them. I
will present a family of elastic metrics and demonstrate its use on
shape space. Following this, I will argue that such metrics are not
always useful, and that the data should inform the choice of connection,
and hence metric, and I will present one possible way to do this.

Thursday, Oct. 4
2:10
Math Dept,. Bahen Centre BA 6290

Klas Modin, University of TorontoOn the geometric approach to image diffusion

I will discuss two geometric approaches to image diffusion, and their
relation to evolution equations on spaces of diffeomorphisms. The
first approach is "Beltrami diffusion", where images are
represented as 2-dimensional submanifolds, called image manifolds,
embedded in a higher dimensional Euclidean space. The Laplace-Beltrami
operator, obtained by pulling back the Euclidean metric, is then used
to smoothen (or sharpen) the image manifold. The second approach is
"hypoelliptic diffusion", in which images are transformed
by subelliptic diffusion equations on the projective tangent bundle
of the plane.